• The Mathematical Education
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• v.52 no.2
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• pp.191-201
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• 2013

The theory of embodied cognition assumes that behaviors, senses and cognitions are closely connected, and there is a growing interest in investigating the significance of embodied cognition in the field of mathematics education. This study aims to applicate the embodied turtle metaphor and expressions when students visualize three-dimensional objects. We used MRT(Verdenberg & Kuse, 1978) & SVT for this research and both tests turned out that turtle schemes are useful to the students in a low level group. In addition, students found turtle schemes more useful in SVT which requires constructing three-dimensional objects, than in MRT which requires just rotating the image of three-dimensional objects in their mind. These results suggest that providing students who are less capable of spatial visualizing with the embodied schemes like turtle metaphor and expressions can be an alternative to improve their spatial visualization ability.

• Journal of English Language & Literature
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• v.59 no.4
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• pp.527-544
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• 2013

William Shakespeare's dirge, "The Phoenix and the Turtle," is still a crux in the Shakespearean canon and interpretation. The poem is still believed a dark allegory dealing with some arcane and obscure courtly matters and politics. However, we cannot recover its allegorical significance. This interpretive situation enforces us to read the poem as a self-conscious artwork in terms of its paradoxical language and meta-poetic metaphors. Paradox, as a subspecies of metaphor, challenges categorical and judgmental absolutes, and produces a sense of wonder in reconciling the logically contradictory opposites. In this poem the urn containing the ashes of the phoenix and the turtle is the icon of the mysterious unity of art, born of the wonderful marriage of male and female. Shakespeare's poem demonstrates in itself the magical power of poetic language in transforming an elegy into an epithalamion. The union of the phoenix and the turtle defies the singularity of their respective entity, and at the same time it retains their distinctive particularity of the two-ness. This neo-Platonic mystery of the "married chastity" is a paradox which confounds reason and verifies the poetic truth of imaginative intellect. The marriage of Christian perichoresis is crystallized in the artwork of the urn, which is admired at by posterity, though the marriage was issueless, due to its passing virtue. "The Phoenix and the Turtle" depicts the metaphor-making process and its effect, the poem.

• Research in Mathematical Education
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• v.15 no.1
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• pp.1-14
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• 2011

Logo has influenced many researchers and learners for the past decades as a 20 turtle geometry environment in the perspective of constructionism. Logo uses the metaphor of 'playing turtle' that is intrinsic, local and procedural. We, then, design an environment in which the metaphor of 'playing turtle' is applied to construct 3D objects, and we figure out ways to represent 3D objects in terms action symbols and 3D building blocks. For this purpose, design three kinds of representation systems, and asked students make various 3D artifacts using various representation systems. We briefly introduce the results of our investigation into students' cognitive burden when they use those representation systems, and discuss the future application measures and the design principles of Logo-based 3D microworld.

• Research in Mathematical Education
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• v.15 no.1
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• pp.15-30
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• 2011

We study to design educational Logo-based microworld environment equipped with 3D construction capability, 3D manipulation, and web-based communication. Extending the turtle metaphor of 2D Logo, we design simple and intuitive symbolic representation system that can create several turtle objects and operations. We also present various mathematization activities applying the turtle objects and suggest the way to make good use of them in mathematics education. In our microworld environment, the symbolic representations constructing the turtle objects can be used for web-based collaborative learning, communication, and assessments.

• Research in Mathematical Education
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• v.16 no.1
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• pp.79-90
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• 2012

This research investigated if the embodied symbol using a turtle metaphor in a microworld environment works as a cognitive tool to mediate the learning of combinatorics. It was found that students were able to not only count the number of cases systematically by using the embodied symbols in a situated problem regarding Permutation and Combination, but also find the rules and infer a concept of Combination through the activities manipulating the symbols. Therefore, we concluded that the embodied symbol, as a bridge that connects learners' concrete experiences with abstract mathematical concepts, can be applied to introduction of various mathematical concepts as well as a combinatorics concept.

• East Asian mathematical journal
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• v.26 no.4
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• pp.447-461
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• 2010

Papert [17] introduced the LOGO environment in which we make a curve using LOGO commands (FORWARD, ROTATE). We call this geometry as turtle geometry. This environment has influenced many researchers and designers of computers and mathematics education. But the curve that we can make using LOGO command is elementary or too difficult. Polygon and circle is elementary and making other curves is difficult. In this paper, we introduce the method of drawing some other curves mediating new command. First, we study epicycloid and hypocycloid in the historical and the physical context. And we introduce the method of making epicycloid and hypocycloid using vector addition. Next we study the polygon patterns of this curve. Finally, we extend the method for making more general curve and we improve the computer environment using this metaphor.